(Almost) Everything You Always Wanted to Know About Deterministic Control Problems in Stratified Domains

TL;DR

This paper revisits and refines the theory of deterministic control problems in stratified domains, improving assumptions, establishing a comparison principle without continuity constraints, and providing a stability framework using PDE methods.

Contribution

It introduces a refined analysis of control problems in stratified domains, extending previous work with weaker assumptions and a broader stability and comparison framework.

Findings

Improved assumptions on dynamics and costs in stratified control problems.

Established a comparison principle for semi-continuous solutions without continuity.

Provided a general stability framework for these control problems.

Abstract

We revisit the pioneering work of Bressan \& Hong on deterministic control problems in stratified domains, i.e. control problems for which the dynamic and the cost may have discontinuities on submanifolds of R N . By using slightly different methods, involving more partial differential equations arguments, we (i) slightly improve the assumptions on the dynamic and the cost; (ii) obtain a comparison result for general semi-continuous sub and supersolutions (without any continuity assumptions on the value function nor on the sub/supersolutions); (iii) provide a general framework in which a stability result holds.